Find the Value Using the Unit Circle cos((4pi)/3)

Problem

cos((4*π)/3)

Solution

1. Identify the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle is in Quadrant III.

2. Determine the reference angle. For an angle θ in Quadrant III, the reference angle θ′ is calculated as θ−π

θ′=(4*π)/3−π=π/3

3. Apply the sign for the cosine function in Quadrant III. In Quadrant III, the xcoordinates are negative, so cos(θ) is negative.

cos((4*π)/3)=−cos(π/3)

4. Evaluate the cosine of the reference angle using known unit circle values.

cos(π/3)=1/2

5. Combine the value and the sign to find the final result.

cos((4*π)/3)=−1/2

Final Answer

cos((4*π)/3)=−1/2

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